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Reynolds transport theorem

(Theorem)

Introduction

Reynolds transport theorem [1] is a fundamental theorem used in formulating the basic laws of fluid mechanics. We will enunciate and demonstrate in this entry the referred theorem. For our purpose, let us consider a fluid flow, characterized by its streamlines, in the Euclidean vector space $(\mathbb{R}^3,\lVert\cdot\rVert)$ and embedded on it we consider, a continuum body $\mathscr{B}$ occupying a volume $\mathscr{V}$ whose particles are fixed by their material (Lagrangian) coordinates $\mathbf{X}$ , and a region $\Re$ where a control volume $\mathfrak{v}$ is defined whose points are fixed by it spatial (Eulerian) coordinates $\mathbf{x}$ and bounded by the control surface $\partial\mathfrak{v}$ . An arbitrary tensor field of any rank is defined over the fluid flow according to the following definition.

Definition We call an extensive tensor property to the expression

$\displaystyle \Psi(\mathbf{x},t):= \int_{\mathfrak{v}}\psi(\mathbf{x},t)\rho(\mathbf{x},t)dv,$

(1)

where $\psi(\mathbf{x},t)$ is the respective intensive tensor property.

Theorem's hypothesis

The kinematics of the continuum can be described by a diffeomorphism $\chi$ which, at any given instant $t\in [0,\infty)\subset\mathbb{R}$ , gives the spatial coordinates $\mathbf{x}$ of the material particle $\mathbf{X}$ ,

$\displaystyle \mathscr{V}\times[0,\infty)\rightarrow \mathfrak{v}\times[0,\infty), \qquad t \mapsto t, \qquad \mathbf{X}\mapsto\mathbf{x}=\chi(\mathbf{X},t).$

Indeed the above sentence corresponds to a change of coordinates which must verify

$\displaystyle J=\bigg\vert\frac{\partial{x}_i}{\partial{X}_j}\bigg\vert\equiv \... ...rt{F_{ij}}\big\vert\neq{0}, \qquad F_{ij}:=\frac{\partial{x}_i}{\partial{X}_j},$

being the Jacobian of transformation and $F_{ij}$ the Cartesian components of the so-called strain gradient tensor $\mathbf{F}$ .

Reynolds transport theorem The material rate of an extensive tensor property associate to a continuum body $\mathscr{B}$ is equal to the local rate of such property in a control volume $\mathfrak{v}$ plus the efflux of the respective intensive property across its control surface $\partial\mathfrak{v}$ .

Proof. By taking on Eq.(1) the material time derivative,

$\displaystyle \frac{D\Psi}{Dt}=\dot{\Psi}=\dot{\overline{\int_{\mathfrak{v}}\ps... ...i\rho{J}}}dV= \int_{\mathscr{V}}(\dot{\overline{\psi\rho}}J+\psi\rho\dot{J})dV=$

$\displaystyle \int_{\mathscr{V}}\Big\{J\Big[\frac{\partial}{\partial{t}}(\psi\r... ...cdot\!\nabla_x(\psi\rho)+ (\psi\rho)\nabla_x\!\cdot\!\mathbf{v}\big]\Big\}(JdV)$

$\displaystyle =\int_{\mathfrak{v}}\frac{\partial}{\partial{t}}(\psi\rho)dv+ \in... ...ho\,dv+ \int_{\partial\mathfrak{v}}\psi\rho\,\mathbf{v}\!\cdot\!\mathbf{n}\,da,$

since $\partial_t(dv)=0$ ( $\mathbf{x}$ fixed) on the first integral and by applying the Gauss-Green divergence theorem on the second integral at the left-hand side. Finally, by substituting Eq.(1) on the first integral at the right-hand side, we obtain

$\displaystyle \dot{\Psi}=\frac{\partial\Psi}{\partial{t}}+ \int_{\partial\mathfrak{v}}\psi\rho\,\mathbf{v}\!\cdot\!\mathbf{n}\,da,$

(2)

endorsing the theorem statement. $\qedsymbol$

Bibliography

1: O. Reynolds, Papers on mechanical and physical subjects-the sub-mechanics of the Universe, Collected Work, Volume III, Cambridge University Press, 1903.

"Reynolds transport theorem" is owned by perucho. [ full author list (2) ]

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Cross-references: side, divergence theorem, integral, derivative, plus, associate, gradient, strain, components, transformation, Jacobian, change of coordinates, sentence, spatial coordinates, diffeomorphism, expression, property, rank, field, tensor, surface, bounded, points, region, coordinates, Lagrangian, fixed, volume, body, continuum, Euclidean vector space, flow
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This is version 4 of Reynolds transport theorem, born on 2006-06-15, modified 2008-04-30.
Object id is 8048, canonical name is ReynoldsTransportTheorem.
Accessed 3634 times total.

Classification:

53A45 (Differential geometry :: Classical differential geometry :: Vector and tensor analysis)

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